Tractability of multivariate analytic problems
Peter Kritzer, Friedrich Pillichshammer, Henryk Wozniakowski
aa r X i v : . [ m a t h . NA ] J u l Tractability of multivariate analytic problems
Peter Kritzer ∗ , Friedrich Pillichshammer † , Henryk Wo´zniakowski ‡ Abstract
In the theory of tractability of multivariate problems one usually studies prob-lems with finite smoothness. Then we want to know which s -variate problems canbe approximated to within ε by using, say, polynomially many in s and ε − functionvalues or arbitrary linear functionals.There is a recent stream of work for multivariate analytic problems for which wewant to answer the usual tractability questions with ε − replaced by 1 + log ε − . Inthis vein of research, multivariate integration and approximation have been studiedover Korobov spaces with exponentially fast decaying Fourier coefficients. This iswork of J. Dick, G. Larcher, and the authors. There is a natural need to analyze moregeneral analytic problems defined over more general spaces and obtain tractabilityresults in terms of s and 1 + log ε − .The goal of this paper is to survey the existing results, present some new results,and propose further questions for the study of tractability of multivariate analyticquestions. Keywords:
Tractability, Korobov space, numerical integration, L -approximation. In this paper we discuss algorithms for multivariate integration or approximation of s -variate functions defined on the unit cube [0 , s . These problems have been studied in alarge number of papers from many different perspectives.The focus of this article is to discuss algorithms for high-dimensional problems definedfor functions from certain Hilbert spaces. There exist many results for such algorithms,and much progress has been made on this subject over the past decades. It is the goalof this review to focus on a recent vein of research that deals with function spaces con-taining analytic periodic functions with exponentially fast decaying Fourier coefficients. ∗ P. Kritzer gratefully acknowledges the support of the Austrian Science Fund, Project P23389-N18and Project F5506-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods:Theory and Applications”. † F. Pillichshammer is supported by the Austrian Science Fund (FWF) Project F5509- N26, which ispart of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. ‡ H. Wo´zniakowski is supported in part by the National Science Foundation.
1e present necessary and sufficient conditions that allow us to obtain exponential errorconvergence and various notions of tractability.We consider algorithms that use finitely many information evaluations. For multivari-ate integration, algorithms use n information evaluations from the class Λ std of standardinformation which consists of only function evaluations. For multivariate approximationin the L -norm, algorithms use n information evaluations either from the class Λ all of allcontinuous linear functionals or from the class Λ std . Since we approximate functions fromthe unit ball of the corresponding space, without loss of generality we restrict ourselves tolinear algorithms that use nonadaptive information evaluations. In all cases, we measurethe error by considering the worst-case error setting. For large s , it is essential to not onlycontrol how the error of an algorithm depends on n , but also how it depends on s . Tothis end, we consider the information complexity, n ( ε, s ), which is the minimal number n for which there exists an algorithm using n information evaluations with an error of atmost ε for the s -variate functions. In all cases considered in this survey, the informationcomplexity is proportional to the minimal cost of computing an ε -approximation sincelinear algorithms are optimal and their implementation cost is proportional to n ( ε, s ).We would like to control how n ( ε, s ) depends on ε − and s . This is the subjectof tractability. In the standard theory of tractability, see [11, 12, 13], weak tractability means that n ( ε, s ) is not exponentially dependent on ε − and s , polynomial tractability means that n ( ε, s ) is polynomially bounded in ε − and s , and strong polynomial tractability means that n ( ε, s ) is polynomially bounded in ε − independently of s .Typically, n ( ε, s ) is polynomially dependent on ε − and s for weighted classes of smoothfunctions. The notion of weighted function classes means that the dependence of func-tions on successive variables and groups of variables is moderated by certain weights. Forsufficiently fast decaying weights, the information complexity depends at most polyno-mially on ε − and s ; hence we obtain polynomial tractability, or even strong polynomial tractability.These notions of tractability are suitable for problems with finite smoothness, that is,when functions from the problem space are differentiable only finitely many times. Thenthe minimal errors e ( n, s ) of algorithms that use n information evaluations typically enjoypolynomial convergence, i.e., e ( n, s ) = O ( n − p ), where the factor in the big O notation aswell as a positive p may depend on s .The case of analytic or infinitely many times differentiable functions is also of interest.For such classes of functions we would like to replace polynomial convergence by expo-nential convergence , and study similar notions of tractability in terms of (1 + log ε − , s )instead of ( ε − , s ). By exponential convergence we mean that e ( n, s ) = O ( q [ O ( n )] p ) with q ∈ (0 , O notation as well as a positive p may depend on s . Exponential convergence with various notions of tractability was studied in the pa-pers [4] and [8] for multivariate integration in weighted Korobov spaces with exponentiallyfast decaying Fourier coefficients. In the paper [2], multivariate L -approximation in theworst-case setting for the same class of functions was considered.In this article, we give an overview of recent results on exponential convergence withdifferent notions of tractability such as weak, polynomial and strong polynomial tractabil-ity in terms of 1+log ε − and s . We also present a few new results and compare conditionswhich are needed for the standard and new tractability notions.In Section 2, we give a short overview of s -variate problems, describe how we measure2rrors, and give precise definitions of various notions of tractability. In Section 3, weintroduce the function class under consideration here, which is a special example of areproducing kernel Hilbert space that was also studied in [2, 4, 8]. In Sections 4 and 5,we provide details on the particular problems of s -variate numerical integration and L -approximation by linear algorithms. We summarize and give an outlook to some relatedopen questions in Section 6. We consider Hilbert spaces H s of s -variate functions defined on [0 , s , and we assumethat there is a family of continuous linear operators S s : H s → G s for s ∈ N , where G s isa normed space.Later, we will introduce a special choice of a Hilbert space H s (cf. Section 3) andstudy two particular examples of s -variate problems, namely: • Numerical integration of functions f ∈ H s , see Section 4. In this case, we have S s ( f ) = R [0 , s f ( x ) d x and G s = R . • L -approximation of functions f ∈ H s , see Section 5. In this case, we have S s ( f ) = f and G s = L ([0 , s ).As already mentioned, without loss of generality, we approximate S s by a linear algo-rithm A n,s using n information evaluations which are given by linear functionals from theclass Λ ∈ { Λ all , Λ std } . That is, A n,s ( f ) = n X j =1 L j ( f ) a j for all f ∈ H s , where L j ∈ Λ and a j ∈ G s for all j = 1 , , . . . , n . For Λ = Λ all we have L j ∈ H ∗ s whereasfor Λ = Λ std we have L j ( f ) = f ( x j ) for all f ∈ H s , and for some x j ∈ [0 , d . For Λ std , wechoose H s as a reproducing kernel Hilbert space so that Λ std ⊂ Λ all .We measure the error of an algorithm A n,s in terms of the worst-case error , which isdefined as e ( H s , A n,s ) := sup f ∈ Hs k f k Hs ≤ k S s ( f ) − A n,s ( f ) k G s , where k·k H s denotes the norm in H s , and k·k G s denotes the norm in G s . The n th minimal(worst-case) error is given by e ( n, s ) := inf A n,s e ( H s , A n,s ) , where the infimum is taken over all admissible algorithms A n,s .For n = 0, we consider algorithms that do not use information evaluations and there-fore we use A ,s ≡
0. The error of A ,s is called the initial (worst-case) error and is givenby e (0 , s ) := sup f ∈ Hs k f k Hs ≤ k S s ( f ) k G s = k S s k . A n,s , we do not only want to control how their errors dependon n , but also how they depend on the dimension s . This is of particular importancefor high-dimensional problems. To this end, we define, for ε ∈ (0 ,
1) and s ∈ N , the information complexity by n ( ε, s ) := min { n : e ( n, s ) ≤ ε } as the minimal number of information evaluations needed to obtain an ε -approximationto S s . In this case, we speak of the absolute error criterion . Alternatively, we can alsodefine the information complexity as n ( ε, s ) := min { n : e ( n, s ) ≤ εe (0 , s ) } , i.e., as the minimal number of information evaluations needed to reduce the initial errorby a factor of ε . In this case we speak of the normalized error criterion .The examples considered in this paper have the convenient property that the initialerrors are one, and the absolute and normalized error criteria coincide. For problemsfor which the initial errors are not one, the results for the absolute and normalized errorcriteria may be quite different; we refer the interested reader to the monographs [11, 12, 13]for further details.The subject of tractability deals with the question how the information complexitydepends on ε − and s . Roughly speaking, tractability means that the information com-plexity lacks a certain disadvantageous dependence on ε − and s .The standard notions of tractability were introduced in such a way that positive resultswere possible for problems with finite smoothness. In this case, one is usually interestedin when n ( ε, s ) depends at most polynomially on ε − and s . The following notions havebeen frequently studied. We say that we have:(a) The curse of dimensionality if there exist positive c, τ and ε such that n ( ε, s ) ≥ c (1 + τ ) s for all ε ≤ ε and infinitely many s. (b) Weak Tractability (WT) iflim s + ε − →∞ log n ( ε, s ) s + ε − = 0 with log 0 = 0 by convention . (c) Polynomial Tractability (PT) if there exist non-negative numbers c, τ , τ such that n ( ε, s ) ≤ c s τ ( ε − ) τ for all s ∈ N , ε ∈ (0 , . (d) Strong Polynomial Tractability (SPT) if there exist non-negative numbers c and τ such that n ( ε, s ) ≤ c ( ε − ) τ for all s ∈ N , ε ∈ (0 , . The exponent τ ∗ of strong polynomial tractability is defined as the infimum of τ forwhich strong polynomial tractability holds.4t turns out that many multivariate problems defined over standard spaces of functionssuffer from the curse of dimensionality. The reason for this negative result is that forstandard spaces all variables and groups of variables are equally important. If we introduceweighted spaces, in which the importance of successive variables and groups of variablesis monitored by corresponding weights, we can vanquish the curse of dimensionality andobtain weak, polynomial or even strong polynomial tractability depending on the decayof the weights. Furthermore, this holds for weighted spaces with finite smoothness. Werefer to [11, 12, 13] for the current state of the art in this field of research.However, the particular weighted function space we are going to define in Section 3 issuch that its elements are infinitely many times differentiable and even analytic. There-fore, it is natural to demand more of the n th minimal errors e ( n, s ) and of the informationcomplexity n ( ε, s ) than for those cases where we only have finite smoothness.To be more precise, we are interested in obtaining exponential or uniform exponentialconvergence of the minimal errors e ( n, s ) for problems with unbounded smoothness. Wenow explain how these notions are defined. By exponential convergence we mean thatthere exist functions q : N → (0 ,
1) and p, C : N → (0 , ∞ ) such that e ( n, s ) ≤ C ( s ) q ( s ) n p ( s ) for all s, n ∈ N . Obviously, the functions q ( · ) and p ( · ) are not uniquely defined. For instance, we can takean arbitrary number q ∈ (0 , C as C ( s ) = (cid:18) log q log q ( s ) (cid:19) /p ( s ) , and then C ( s ) q ( s ) n p ( s ) = C ( s ) q ( n/C ( s )) p ( s ) . We prefer to work with the latter bound which was also considered in [2, 8].We say that we achieve exponential convergence (EXP) for e ( n, s ) if there exist anumber q ∈ (0 ,
1) and functions p, C, C : N → (0 , ∞ ) such that e ( n, s ) ≤ C ( s ) q ( n/C ( s )) p ( s ) for all s, n ∈ N . (1)If (1) holds we would like to find the largest possible rate p ( s ) of exponential convergencedefined as p ∗ ( s ) = sup { p ( s ) : p ( s ) satisfies (1) } . We say that we achieve uniform exponential convergence (UEXP) for e ( n, s ) if thefunction p in (1) can be taken as a constant function, i.e., p ( s ) = p > s ∈ N .Similarly, let p ∗ = sup { p : p ( s ) = p > s ∈ N } denote the largest rate of uniform exponential convergence.Exponential convergence implies that asymptotically, with respect to ε tending tozero, we need O (log /p ( s ) ε − ) information evaluations to compute an ε -approximation.However, it is not clear how long we have to wait to see this nice asymptotic behaviorespecially for large s . This, of course, depends on how C ( s ) , C ( s ) and p ( s ) depend5n s , and it is therefore near at hand to adapt the concepts (b)–(d) of tractability toexponential error convergence. Indeed, we would like to replace ε − by 1 + log ε − inthe standard notions (b)–(d), which yields new versions of weak, polynomial, and strongpolynomial tractability. The following new tractability versions (e), (f), and (g) werealready introduced in [2, 4, 8]. We use a new kind of notation in order to be able todistinguish (b)–(d) from (e)–(g). We say that we have:(e) Exponential Convergence-Weak Tractability (EC-WT) iflim s +log ε − →∞ log n ( ε, s ) s + log ε − = 0 with log 0 = 0 by convention . (f) Exponential Convergence-Polynomial Tractability (EC-PT) if there exist non-nega-tive numbers c, τ , τ such that n ( ε, s ) ≤ c s τ (1 + log ε − ) τ for all s ∈ N , ε ∈ (0 , . (g) Exponential Convergence-Strong Polynomial Tractability (EC-SPT) if there existnon-negative numbers c and τ such that n ( ε, s ) ≤ c (1 + log ε − ) τ for all s ∈ N , ε ∈ (0 , . The exponent τ ∗ of EC-SPT is defined as the infimum of τ for which EC-SPT holds.Let us give some comments on these definitions. First, we remark that the use of theprefix EC (exponential convergence) in (e)–(g) is motivated by the fact that EC-PT (andtherefore also EC-SPT) implies exponential convergence (cf. Theorem 3). Also EC-WTimplies that e ( n, s ) converges to zero faster than any power of n − as n goes to infinity,i.e., for any α > n →∞ n α e ( n, s ) = 0 . (2)This can be seen as follows. Let α > δ ∈ (0 , α ). For a fixed dimension s , EC-WT implies the existence of an M = M ( δ ) > ε > ε − > M we have log n ( ε, s )log ε − < δ ⇔ n ( ε, s ) < ε − δ . This implies that for large enough n ∈ N we have e ( n, s ) < n − /δ . Hence, we have n α e ( n, s ) < n α − /δ → n → ∞ .Furthermore we note, as in [2, 4], that if (1) holds then n ( ε, s ) ≤ & C ( s ) (cid:18) log C ( s ) + log ε − log q − (cid:19) /p ( s ) ' for all s ∈ N and ε ∈ (0 , . (3)Moreover, if (3) holds then e ( n + 1 , s ) ≤ C ( s ) q ( n/C ( s )) p ( s ) for all s, n ∈ N . This means that (1) and (3) are practically equivalent. Note that 1 /p ( s ) determines thepower of log ε − in the information complexity, whereas log q − affects only the multiplierof log /p ( s ) ε − . From this point of view, p ( s ) is more important than q .6n particular, EC-WT means that we rule out the cases for which n ( ε, s ) dependsexponentially on s and log ε − .For instance, assume that (1) holds. Then uniform exponential convergence (UEXP)implies EC-WT if C ( s ) = exp (exp ( o ( s ))) and C ( s ) = exp( o ( s )) as s → ∞ . These conditions are rather weak since C ( s ) can be almost doubly exponential and C ( s )almost exponential in s .The definition of EC-PT (and EC-SPT) implies that we have uniform exponentialconvergence with C ( s ) = e (where e denotes exp(1)), q = 1 / e, C ( s ) = c s τ and p = 1 /τ .Obviously, EC-SPT implies C ( s ) = c and τ ∗ ≤ /p ∗ .If (3) holds then we have EC-PT if p := inf s p ( s ) > A, A and η, η such that C ( s ) ≤ exp ( As η ) and C ( s ) ≤ A s η for all s ∈ N . The condition on C ( s ) seems to be quite weak since even for singly exponential C ( s ) wehave EC-PT. Then τ = η + η/p and τ = 1 /p . EC-SPT holds if C ( s ) and C ( s ) areuniformly bounded in s , and then τ ∗ ≤ /p .We briefly mention a recent paper [14], where a new notion of weak tractability isdefined similarly to EC-WT. Namely, let κ ≥
1. Then it is required thatlim s +log ε − →∞ log n ( ε, s ) s + [log ε − ] κ = 0 with log 0 = 0 by convention . (4)Obviously, for κ = 1 this is the same as EC-WT. However, for κ > H s and study theproblems of s -variate integration and L -approximation. In this article, we choose for the Hilbert space H s a weighted Korobov space of periodicand smooth functions, which is probably the most popular kind of space used to analyzeperiodic functions. Such Korobov spaces can be defined via a reproducing kernel (forgeneral information on reproducing kernel Hilbert spaces, see [1]) of the form K s ( x , y ) = X h ∈ Z s ρ h exp(2 π i h · ( x − y )) for all x , y ∈ [0 , s (5)with the usual dot product h · ( x − y ) = s X j =1 h j ( x j − y j ) , where h j , x j , y j are the j th components of the vectors h , x , y , respectively. Furthermore, i = √−
1. The nonnegative ρ h for h ∈ Z s , which may also depend on s and other7arameters, are chosen such that P h ∈ Z s ρ h < ∞ . This choice guarantees that the kernel K s is well defined, since | K s ( x , y ) | ≤ K s ( x , x ) = X h ∈ Z s ρ h < ∞ . Obviously, the function K s is symmetric in x and y and it is easy to show that it is alsopositive definite. Therefore, K s ( x , y ) is indeed a reproducing kernel. The correspondingKorobov space is denoted by H ( K s ).The smoothness of the functions from H ( K s ) is determined by the decay of the ρ h ’s.A very well studied case in literature is for Korobov spaces of finite smoothness α . Here ρ h is of the form ρ h = r α, γ ( h ) , where α > γ = ( γ , γ , . . . ) is a sequence of positive reals, and for h =( h , . . . , h s ) we have r α, γ ( h ) = s Y j =1 r α,γ j ( h j ) , with r α,γ (0) = 1 and r α,γ ( h ) = γ | h | − α whenever h = 0.Hence the ρ h ’s decay polynomially in the components of h . The parameter α guar-antees the existence of some partial derivatives of the functions and the so-called weights γ model the influence of the different components on the variation of the functions fromthe Korobov space. More information can be found in [11, Appendix A.1].The idea of introducing weights stems from Sloan and Wo´zniakowski and was firstdiscussed in [16]. For multivariate integration defined over weighted Korobov spaces ofsmoothness α , algorithms based on n function evaluations can obtain the best possibleconvergence rate of order O ( n − α/ δ ) for any δ >
0. Under certain conditions on theweights, weak, polynomial or even strong polynomial tractability in the sense of (b)–(d)can be achieved. We refer to [11, 12, 13] and the references therein and to the recentsurvey [3] for further details.Besides the case of finite smoothness, Korobov spaces of infinite smoothness werealso considered. In this case, the ρ h ’s decay to zero exponentially fast in h . Multivariateintegration and L -approximation for such Korobov spaces have been analyzed in [2, 4, 8].To model the influence of different components we use two weight sequences a = { a j } j ≥ and b = { b j } j ≥ . In order to guarantee that the kernel that we will introduce in a moment is well definedwe must assume that a j > b j >
0. In fact, we assume a little more throughout thepaper, namely that with the proper ordering of variables we have0 < a ≤ a ≤ · · · and b ∗ = inf b j > . (6)Let a ∗ = inf a j which is a in our case.Fix ω ∈ (0 ,
1) and put in (5) ρ h = ω h := ω P sj =1 a j | h j | bj for all h = ( h , h , . . . , h s ) ∈ Z s . (7)8or this choice of ρ h we denote the kernel in (5) by K s, a , b . We suppress the dependence on ω in the notation since ω will be fixed throughout the paper and a and b will be varied.Note that K s, a , b is well defined since X h ∈ Z s ω h = s Y j =1 ∞ X h =1 ω a j h bj ! ≤ ∞ X h =1 ω a ∗ h b ∗ ! s < ∞ . The last series is finite by the comparison test because a ∗ > b ∗ > K s, a , b is denoted by H ( K s, a , b ). Clearly,functions from H ( K s, a , b ) are infinitely many times differentiable, see [4], and they areeven analytic as shown in [2, Proposition 2].For f ∈ H ( K s, a , b ) we have f ( x ) = X h ∈ Z s b f ( h ) exp(2 π i h · x ) for all x ∈ [0 , s , where b f ( h ) = R [0 , s f ( x ) exp( − π i h · x ) d x is the h th Fourier coefficient of f . The innerproduct of f and g from H ( K s, a , b ) is given by h f, g i H ( K s, a , b ) = X h ∈ Z s b f ( h ) b g ( h ) ω − h and the norm of f from H ( K s, a , b ) by k f k H ( K s, a , b ) = X h ∈ Z s | b f ( h ) | ω − h ! / < ∞ . Define the functions e h ( x ) = exp(2 π i h · x ) ω / h for all x ∈ [0 , s . (8)Then { e h } h ∈ Z s is a complete orthonormal basis of the Korobov space H ( K s, a , b ). H ( K s, a , b ) In this section we study numerical integration, i.e., we are interested in numerical approx-imation of the values of integrals I s ( f ) = Z [0 , s f ( x ) d x for all f ∈ H ( K s, a , b ) . Using the general notation from Section 2, we now have S s ( f ) = I s ( f ) for functions f ∈ H s = H ( K s, a , b ), and G s = C .We approximate I s ( f ) by means of linear algorithms Q n,s of the form Q n,s ( f ) := n X k =1 q k f ( x k ) , q k ∈ C and sample points x k ∈ [0 , s . If we choose q k = 1 /n for all k = 1 , , . . . , n then we obtain so-called quasi-Monte Carlo (QMC) algorithms which areoften used in practical applications especially if s is large. For recent overviews of thestudy of QMC algorithms we refer to [3, 5, 9].The n th minimal worst-case error is given by e int ( n, s ) = inf q k , x k , k =1 , ,...,n sup f ∈ H ( Ks, a , b ) k f k H ( Ks, a , b ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) I s ( f ) − n X k =1 q k f ( x k ) (cid:12)(cid:12)(cid:12)(cid:12) . It is well known, see for instance [12, 17], that e int ( n, s ) = inf x k , k =1 , ,...,n sup f ∈ H ( Ks, a , b ) , f ( x k )=0 , k =1 , ,...,n k f k H ( Ks, a , b ) ≤ | I s ( f ) | . (9)For n = 0, the best we can do is to approximate I s ( f ) simply by zero, and e int (0 , s ) = k I s k = 1 for all s ∈ N . Hence, the integration problem is well normalized for all s .We now summarize the main results regarding numerical integration in H ( K s, a , b ).Here and in the following, we will be using the notational abbreviationsEXP UEXPWT PT SPTEC-WT EC-PT EC-SPTto denote exponential and uniform exponential convergence, and weak, polynomial andstrong polynomial tractability in terms of (b)–(d) and (e)–(g). We now state relationsbetween these concepts as well as necessary and sufficient conditions on a and b for whichthese concepts hold. As we shall see, in the settings considered in this paper, manyconditions for obtaining these concepts are equivalent.We first state a theorem which describes conditions on the weight sequences a and b toobtain exponential (EXP) and uniform exponential (UEXP) convergence. This theoremis from [2, 8]. Theorem 1
Consider integration defined over the Korobov space H ( K s, a , b ) with weightsequences a and b satisfying (6) . • EXP holds for all considered a and b and p ∗ ( s ) = 1 B ( s ) with B ( s ) := s X j =1 b j . • UEXP holds iff b is such that B := ∞ X j =1 b j < ∞ . If so then p ∗ = 1 /B . b j go toinfinity so fast that B := P ∞ j =1 b − j < ∞ , with no extra conditions on a j and ω . The largestexponent p of uniform exponential convergence is 1 /B . Hence for small B the exponent p is large. For instance, for b j = j − we have B = π / p ∗ = 6 /π = 0 . . . . .Next, we consider standard notions of tractability, (b)–(d). They have not yet beenstudied for the Korobov space H ( K s, a , b ) and therefore we need to prove the next theorem. Theorem 2
Consider integration defined over the Korobov space H ( K s, a , b ) with weightsequences a and b satisfying (6) . For simplicity, assume that A := lim j →∞ a j log j exists. • SPT holds if
A > ω − . In this case the exponent τ ∗ of SPT satisfies τ ∗ ≤ min (cid:18) , A log ω − (cid:18) A log ω − (cid:19)(cid:19) . On the other hand, if we have SPT with exponent τ ∗ , then A ≥ τ ∗ log ω − . • PT holds if there is an integer j ≥ such that a j log j ≥ ω − for all j ≥ j . • WT holds if lim j →∞ a j = ∞ .Proof. It is well known that integration is no harder than L -approximation for the classΛ std . For the Korobov class the initial errors of integration and approximation are 1.Therefore the corresponding notions of tractability for approximation imply the samenotions of tractability for integration. From Theorem 5, presented in the next section, wethus conclude SPT, PT and WT also for integration. The second bound on the exponent τ ∗ of SPT also follows from Theorem 5. It remains to prove that τ ∗ ≤
2. It is known, see,e.g., [12, Theorem 10.4], that[ e int ( n, s )] ≤ n Z [0 , s K s, a , b ( x , x ) d x ≤ n s Y j =1 ∞ X h =1 ω a j h b ∗ ! . It is shown in the proof of Theorem 5 (with τ = 1) that A > C ∈ (0 , ∞ ) such that 2 P ∞ h =1 ω a j h b ∗ ≤ C ω a j . Therefore[ e int ( n, s )] ≤ n s Y j =1 C ∞ X j =1 ω a j ! ≤ n exp C s X j =1 ω a j ! . Note that for j ≥ ω a j = j − a j (log ω − ) / log j . Since A > / (log ω − ) for large j weconclude that ω a j ≤ j − β with β ∈ (1 , A log ω − ). Hence P ∞ j =1 ω a j < ∞ and e ( n, s ) ≤ ε n = O ( ε − ) with the factor in the big O notation independent of s . This implies SPTwith the exponent at most 2.It remains to show the necessary condition for SPT with exponent τ ∗ . First of all weshow the estimate e int ( s, s ) ≥ ω a s √ ω a s for all s ∈ N . (10)Let h (0) = (0 , , . . . , ∈ Z s . For j = 1 , , . . . , s , let h ( j ) = (0 , , . . . , , , . . . , ∈ Z s with 1 on the j th place . For h ∈ Z s , let c h ( x ) = exp π i s X j =1 h j x j ! for all x ∈ [0 , s . For j = 0 , , . . . , s , note that c h ( j ) ( x ) = exp(2 π i x j ) and c h ( j ) ( x ) c h ( k ) ( x ) = c h ( j ) − h ( k ) ( x ) . Consider the function f ( x ) = s X j =0 α j c h ( j ) ( x ) for all x ∈ [0 , s for some complex numbers α j .We know that adaption does not help for the integration problem. Suppose that wesample functions at s nonadaptive points x , x , . . . , x s ∈ [0 , s . We choose numbers α j such that f ( x j ) = 0 for all j = 1 , , . . . , s. This corresponds to s homogeneous linear equations in ( s + 1) unknowns. Therefore thereexists a nonzero solution α , α , . . . , α s which we may normalize such that s X j =0 | α j | = 1 . Let g ( x ) = f ( x ) f ( x ) = s X j,k =0 α j α k c h ( j ) − h ( k ) ( x ) for all x ∈ [0 , s . Clearly, g ( x j ) = 0 for all j = 1 , , . . . , s . Since I s ( c h ( j ) − h ( k ) ) = 0 for j = k and 1 for j = k we obtain I s ( g ) = s X j =0 | α j | = 1 . Now it follows from (9) that e int ( s, s ) ≥ I s (cid:18) g k g k H ( K s, a , b ) (cid:19) = 1 k g k H ( K s, a , b ) . g from above. Note that k g k H ( K s, a , b ) = h g, g i H ( K s, a , b ) = * s X j ,k =0 α j α k c h ( j − h ( k , s X j ,k =0 α j α k c h ( j − h ( k + H ( K s, a , b ) = s X j ,k ,j ,k =0 α j α j α k α k (cid:10) c h ( j − h ( k , c h ( j − h ( k (cid:11) H ( K s, a , b ) . For h ( j ) − h ( k ) = h ( j ) − h ( k ) we have (cid:10) c h ( j − h ( k , c h ( j − h ( k (cid:11) H ( K s, a , b ) = 0 , whereas for h ( j ) − h ( k ) = h ( j ) − h ( k ) we have (cid:10) c h ( j − h ( k , c h ( j − h ( k (cid:11) H ( K s, a , b ) = ω − h ( j − h ( k . Therefore it is enough to consider h ( j ) − h ( k ) = h ( j ) − h ( k ) . Suppose first that j = k . Then h ( j ) − h ( k ) = h ( j ) − h ( k ) implies that j = j and k = k and ω − h ( j − h ( k = ω − a j − a k . On the other hand, if j = k then h ( j ) − h ( k ) = h (0) which implies that j = k and ω − h ( j − h ( k = 1 . Therefore k g k H ( K s, a , b ) = s X j ,k ,j ,k =0 , h ( j − h ( k = h ( j − h ( k α j α j α k α k ω h ( j − h ( k = s X j =0 s X k =0 ,k = j | α j | | α k | ω − a j − a k + s X j =0 | α j | s X j =0 | α j | = s X j =0 | α j | ω − a j −| α j | ω − a j + s X k =0 | α k | ω − a k ! + 1 ≤ s X j =0 | α j | ω − a j ! + 1 ≤ ω − a s + 1 . Hence, k g k H ( K s, a , b ) ≤ √ ω − a s = √ ω a s ω a s . Finally, e int ( s, s ) ≥ k g k H ( K s, a , b ) ≥ ω a s √ ω a s , τ ∗ . This means that for any positive δ there exists a positive number C δ such that n ( ε, s ) ≤ C δ ε − ( τ ∗ + δ ) for all ε ∈ (0 , , s ∈ N . Let n = n ( ε ) := ⌊ C δ ε − ( τ ∗ + δ ) ⌋ . Then e int ( n ( ε ) , s ) ≤ ε for all s ∈ N . Taking s = n ( ε ), we conclude from (10) that ω a s √ ω a s ≤ e int ( s, s ) ≤ ε, which implies (1 − ε ) ω a s ≤ ε . Taking logarithms this means that a s log ε − ≥ o (1)log ω − as ε → . Since log ε − = (1 + o (1))( τ ∗ + δ ) − log s we finally have A = lim s →∞ a s log s ≥ τ ∗ + δ ) log ω − . Since δ can be arbitrarily small, the proof is completed. ✷ We stress that for integration we only know sufficient conditions on a and b for thestandard notions PT and WT. Obviously, it would be welcome to find also necessaryconditions and verify if they match the conditions presented in the last theorem. ForSPT we have a sufficient condition and a necessary condition, but there remains a (small)gap between these. Again, it would be welcome to find matching sufficient and necessaryconditions for SPT. Note that it may happen that A = ∞ . This happens when a j ’s goto infinity faster than log j . In this case, the exponent of SPT is zero. This means thatfor any positive δ , no matter how small, n ( ε, s ) = O ( ε − δ ) with the factor in the big O notation independent of s . We also stress that the conditions on all standard notions oftractability depend only on a and are independent of b .Finally, we have a result regarding the EC notions of tractability, (d)–(f). The subse-quent theorem follows by combining the findings in [8] and [2, Section 9]. Theorem 3
Consider integration defined over the Korobov space H ( K s, a , b ) with weightsequences a and b satisfying (6) . Then the following results hold: • EC-PT (and, of course, EC-SPT) implies UEXP. • We have EC-WT ⇔ lim j →∞ a j = ∞ , EC-WT+UEXP ⇔ B < ∞ and lim j →∞ a j = ∞ . The following notions are equivalent:EC-PT ⇔ EC-PT+EXP ⇔ EC-PT+UEXP ⇔ EC-SPT ⇔ EC-SPT+EXP ⇔ EC-SPT+UEXP . • EC-SPT+UEXP holds iff b − j ’s are summable and a j ’s are exponentially large in j ,i.e., B := ∞ X j =1 b j < ∞ and α ∗ := lim inf j →∞ log a j j > . Then the exponent τ ∗ of EC-SPT satisfies τ ∗ ∈ (cid:20) B, B + min (cid:18) B, log 3 α ∗ (cid:19)(cid:21) . In particular, if α ∗ = ∞ then τ ∗ = B . Theorem 3 states that EC-PT implies UEXP and hence
B < ∞ . The notion of EC-PTis therefore stronger than the notion of uniform exponential convergence. EC-WT holdsif and only if the a j ’s tend to infinity. This holds independently of the weights b andindependently of the rate of convergence of a to infinity. As already shown, this impliesthat (2) holds. Furthermore, EC-WT+UEXP holds if additionally B < ∞ . Hence forlim j a j = ∞ and B = ∞ , EC-WT holds without UEXP. It is a bit surprising that thenotions of EC-tractability with uniform exponential convergence are equivalent. Necessaryand sufficient conditions for EC-SPT with uniform exponential convergence are B < ∞ and α ∗ >
0. The last condition means that a j ’s are exponentially large in j for large j . L -approximation in H ( K s, a , b ) Let us now turn to approximation in the space H ( K s, a , b ). We study L -approximation offunctions from H ( K s, a , b ). This problem is defined as an approximation of the embeddingfrom the space H ( K s, a , b ) to the space L ([0 , s ), i.e.,EMB s : H ( K s, a , b ) → L ([0 , s ) given by EMB s ( f ) = f. In terms of the notation in Section 2, S s ( f ) = EMB s ( f ) = f for f ∈ H ( K s, a , b ), and G s = L ([0 , s ).Without loss of generality, see again [11, 17], we approximate EMB s by linear algo-rithms A n,s of the form A n,s ( f ) = n X k =1 α k L k ( f ) for f ∈ H ( K s, a , b ) , (11)where each α k is a function from L ([0 , s ) and each L k is a continuous linear functionaldefined on H s from a permissible class Λ of information, Λ ∈ { Λ all , Λ std } . Since H ( K s, a , b ) isa reproducing kernel Hilbert space, function evaluations are continuous linear functionalsand therefore Λ std ⊆ Λ all . 15et e L − app , Λ ( n, s ) be the n th minimal worst-case error, e L − app , Λ ( n, s ) = inf A n,s e L − app ( H ( K s, a , b ) , A n,s ) , where the infimum is taken over all linear algorithms A n,s of the form (11) using informa-tion from the class Λ ∈ { Λ all , Λ std } . For n = 0 we simply approximate f by zero, and theinitial error is e L − app , Λ (0 , s ) = k EMB s k = sup f ∈ H ( Ks, a , b ) k f k H ( Ks, a , b ) ≤ k f k L ([0 , s ) = 1 . This means that also L -approximation is well normalized for all s ∈ N .Let us now outline the main results regarding L -approximation in H ( K s, a , b ). Again,we start with results on EXP and UEXP. The following result was proved in [2]. Theorem 4
Consider L -approximation defined over the Korobov space H ( K s, a , b ) withweight sequences a and b satisfying (6) . Then the following results hold for both classes Λ all and Λ std : • EXP holds for all considered a and b with p ∗ ( s ) = 1 B ( s ) with B ( s ) := s X j =1 b j . • UEXP holds iff a is an arbitrary sequence and b is such that B := ∞ X j =1 b j < ∞ . If so then p ∗ = 1 /B . Note that the conditions are the same as for the integration problem in Theorem 1.Hence the comments following Theorem 1 also apply for approximation. Beyond that itis interesting that we have the same conditions for Λ all and Λ std , although the class Λ std is much smaller than the class Λ all .We now address conditions on the weights a and b for the standard concepts oftractability. This has not yet been done before for ω h of the form (7), and thereforewe need to prove the next theorem. Theorem 5
Consider L -approximation defined over the Korobov space H ( K s, a , b ) witharbitrary sequences a and b satisfying (6) . Assume for simplicity that A := lim j →∞ a j log j exists. Then the following results hold:For Λ all we have: SPT ⇔ A > . In this case, the exponent of SPT is [ τ all ] ∗ = 2 A log ω − . • PT ⇔ SPT . • WT holds for all considered a and b .For Λ std we have: • SPT holds if
A > / (log ω − ) . In this case, the exponent [ τ std ] ∗ satisfies [ τ all ] ∗ ≤ [ τ std ] ∗ ≤ [ τ all ] ∗ + 12 ([ τ all ] ∗ ) < [ τ all ] ∗ + 2 . On the other hand, if we have SPT with exponent [ τ std ] ∗ , then A ≥ τ std ] ∗ log ω − . • PT holds if there is an integer j ≥ such that a j log j ≥ ω − for all j ≥ j . • WT holds if lim j →∞ a j = ∞ .Proof. Consider first the class Λ all . • From [11, Theorem 5.2] it follows that SPT for Λ all is equivalent to the existence ofa number τ > C SPT ,τ := sup s X h ∈ Z s ω τ h ! /τ < ∞ . Note that X h ∈ Z s ω τ h = s Y j =1 ∞ X h =1 ω τa j h bj ! = s Y j =1 ω τa j ∞ X h =1 ω τa j ( h bj − ! . We have 1 ≤ ∞ X h =1 ω τa j ( h bj − ≤ ∞ X h =1 ω τa ∗ ( h b ∗ − =: A τ . We can rewrite A τ as A τ = ∞ X h =1 h − x h , where x = 1 and for h ≥ x h = τ a ∗ (log ω − ) h b ∗ − h . h x h = ∞ the last series is convergent and therefore A τ < ∞ . This provesthat X h ∈ Z s ω τ h = s Y j =1 (1 + 2 A ( τ ) ω τa j ) with A ( τ ) ∈ [1 , A τ ] . This implies thatsup s X h ∈ Z s ω τ h ! /τ = ∞ Y j =1 (1 + 2 A ( τ ) ω τa j ) /τ < ∞ iff ∞ X j =1 ω τa j < ∞ . We now show that ∞ X j =1 ω τa j < ∞ for some τ iff A > . Indeed, for j ≥ ω τa j = j − y j with y j = τ log ω − a j log j . If A > δ we can choose τ such that y j ≥ δ forsufficiently large j and therefore the series ∞ X j =1 ω τa j = ω τa + ∞ X j =2 j − y j is convergent.If A = 0 then independently of τ the series P ∞ j =1 ω τa j is divergent. Indeed, thenlim j y j = 0 and for an arbitrary positive δ ≤ τ we can choose j ( δ, τ ) such that y j ∈ (0 , δ ) for all j ≥ j ( δ, τ ) and ∞ X j =1 ω τa j ≥ ∞ X j = j ( δ,τ ) j − δ = ∞ , as claimed. This proves that SPT holds iff A > τ ∗ , where τ ∗ isthe infimum of τ for which C SPT ,τ < ∞ . In our case, it is clear that we must have τ ≥ (1 + δ ) / (( A − δ ) log ω − ) for arbitrary δ ∈ (0 , A ). This completes the proof ofthis point. • To show that PT is equivalent to SPT, it is obviously enough to show that PT impliesSPT. According to [11, Theorem 5.2], PT for Λ all is equivalent to the existence ofnumbers τ > q ≥ C PT := sup s X h ∈ Z s ω τ h ! /τ s − q < ∞ . This means that log X h ∈ Z s ω τ h ≤ τ (log C PT + q log s ) . (12)18rom the previous considerations we know thatlog X h ∈ Z s ω τ h = log s Y j =1 (1 + 2 A ( τ ) ω τa j ) = s X j =1 log (1 + 2 A ( τ ) ω τa j ) . Assume that A = 0. Suppose first that a j ’s are uniformly bounded. Thenlog P h ∈ Z s ω τ h is of order s which contradicts the inequality (12). Assume nowthat lim j a j = ∞ . Then log P h ∈ Z s ω τ h is of order P sj =2 ω τa j = P sj =2 j − y j . Sincelim j y j = 0 we have for δ ∈ (0 , j − y j ≥ j − δ for large j . This proves that P sj =2 j − δ ≈ R s x − δ d x is of order s − δ which again contradicts the inequality (12).Hence, A > • We now show WT for all a and b with a ∗ , b ∗ >
0. We have ω h = ω P sj =1 a j | h j | bj ≤ ω ∗ , h := ω a ∗ P sj =1 | h j | b ∗ . Note that for h = we have ω h = ω ∗ , h = 1. This shows that the approximationproblem with ω h is not harder than the approximation problem with ω ∗ , h . Thelatter problem is a linear tensor product problem with the univariate eigenvalues of W = EMB ∗ EMB : H ( K ,a ∗ ,b ∗ ) → H ( K ,a ∗ ,b ∗ ) given by λ = 1 , λ j = λ j +1 = ω a ∗ j b ∗ for all j ≥ . Clearly, λ < λ and λ j goes to zero faster than polynomially with j . This impliesWT due to [11, Theorem 5.5]. We now turn to the class Λ std . • For
A > / (log ω − ) we have [ τ all ] ∗ <
2. From [13, Theorem 26.20] we get SPTfor Λ std as well as the bounds on [ τ std ] ∗ . The necessary condition for SPT withexponent [ τ std ] ∗ follows from Theorem 2. • To obtain PT we use [13, Theorem 26.13] which states that polynomial tractabilitiesfor Λ std and Λ all are equivalent if trace( W s ) = O ( s q ) for some q ≥
0, where trace( W s )is the sum of the eigenvalues of the operator W s = EMB ∗ s EMB s : H ( K s, a , b ) → H ( K s, a , b ) . In our case, W s is given by W s f = X h ∈ Z s ω h h f, e h i H ( K s, a , b ) e h with e h given by (8). The eigenpairs of W s are ( ω h , e h ) since W s e h = ω h e h = ω P sj =1 a j | h j | bj e h for all h ∈ Z s In fact, we also have quasi-polynomial tractability, i.e., n ( ε, s ) ≤ C exp( t (1 + log ε − )(1 + log s )) forsome C > t ≈ / ( a ∗ log ω − ), see [7]. W s ) = s Y j =1 (1 + 2 A (1) ω a j ) ≤ exp A (1) s X j =1 ω a j ! . Due to the assumption a j / log j ≥ / (log ω − ) for j ≥ j we have ω a j ≤ j − for j ≥ j . Therefore there is a positive C such thattrace( W s ) ≤ C exp A (1) s X j = j j − ! ≤ C s A (1) . This proves that PT for Λ std holds iff PT for Λ all holds. As we already proved, thelatter holds iff
A >
0. The assumption on a j implies that A ≥ / (log ω − ) > • To obtain WT we use [13, Theorem 26.11]. This theorem states that weak tractabil-ities for classes Λ std and Λ all are equivalent if log trace( W s ) = o ( s ). The proof ofTheorem 4 in [2] yields that lim j a j = ∞ implies P sj =1 ω a j = o ( s ). Hence,log trace( W s ) ≤ log (exp (2 A (1) o ( s ))) = o ( s ) , as needed. ✷ We briefly comment on Theorem 5. For the class Λ all we know necessary and sufficientconditions on SPT, PT and WT if the limit of a j / log j exists. It is interesting to study thecase when the last limit does not exist. It is easy to check that A inf := lim inf j a j / log j > A inf > std we only know sufficient conditions for PT and WT. It would be ofinterest to verify if these conditions are also necessary. For SPT, as for multivariate inte-gration, there remains a (small) gap between sufficient and necessary conditions. Againit would be desirable to close this gap.Finally, we have results regarding the EC-notions of tractability, (e)–(g). The subse-quent theorem has been shown in [2]. Theorem 6
Consider L -approximation defined over the Korobov space H ( K s, a , b ) witharbitrary sequences a and b satisfying (6) . Then the following results hold for both classes Λ all and Λ std : • EC-PT (and, of course, EC-SPT) tractability implies uniform exponential conver-gence, EC-PT ⇒ UEXP . • We have EC-WT ⇔ lim j →∞ a j = ∞ , EC-WT+UEXP ⇔ B < ∞ and lim j →∞ a j = ∞ . The following notions are equivalent:EC-PT ⇔ EC-PT+EXP ⇔ EC-PT+UEXP ⇔ EC-SPT ⇔ EC-SPT+EXP ⇔ EC-SPT+UEXP . • EC-SPT+UEXP holds iff b − j ’s are summable and a j ’s are exponentially large in j ,i.e., EC-SPT+UEXP ⇔ B := ∞ X j =1 b j < ∞ and α ∗ := lim inf j →∞ log a j j > . Then the exponent τ ∗ of EC-SPT satisfies τ ∗ ∈ (cid:20) B, B + min (cid:18) B, log 3 α ∗ (cid:19)(cid:21) . In particular, if α ∗ = ∞ then τ ∗ = B . Again, the conditions are the same as for the integration problem in Theorem 3 andwe have the same conditions for Λ all and Λ std . The comments following Theorem 3 applyalso for approximation. We remark that the results are constructive. The correspondingalgorithms for the class Λ all and Λ std can be found in [2].We want to stress that for the class Λ std we obtain the results of Theorem 6 bycomputing function values at grid points with varying mesh-sizes for successive variables.Such grids are also successfully used for multivariate integration in [2, 8]. This relativelysimple design of sample points should be compared with the design of (almost) optimalsample points for analogue problems defined over spaces of finite smoothness. In thiscase, the design is much harder and requires the use of deep theory of digital nets andlow discrepancy points, see [5, 10].It is worth adding that if we use the definition (4) of WT with κ > /b ∗ then it isproved in [14] that WT holds even for a j = a > b j = b ∗ >
0. Hence, the conditionlim j a j = ∞ which is necessary and sufficient for EC-WT is now not needed. The study of tractability with exponential convergence is a new research subject. Wepresented a handful of results only for multivariate integration and approximation prob-lems defined over Korobov spaces of analytic functions. Obviously, such a study shouldbe performed for more general multivariate problems defined over more general spaces of C ∞ or analytic functions. It would be very much desirable to characterize multivariateproblems for which various notions of tractability with exponential convergence hold. Inthis survey we presented the notions of EC-WT, EC-PT and EC-SPT. We believe thatother notions of tractability with exponential convergence should be also studied. In fact,all notions which were presented for tractability with respect to the pairs ( ε − , s ) can beeasily generalized and studied for the pairs (1 + log ε − , s ). In particular, the notions ofEC-QPT (exponential convergence-quasi polynomial tractability) and EC-UWT (expo-nential convergence-uniform weak tractability) are probably the first candidates for such21 study. Quasi-polynomial tractability was briefly mentioned in the footnote of Section 5.Uniform weak tractability generalizes the notion of weak tractability and means that n ( ε, s ) is not exponential in ε − α and s β for all positive α and β , see [15].The proof technique used for EC-tractability of integration and approximation is quitedifferent than the proof technique used for standard tractability. Furthermore, it seemsthat some results are easier to prove for EC-tractability than their counterparts for thestandard tractability. In particular, optimal design of sample points seems to be such anexample. We are not sure if this holds for other multivariate problems.We hope that exponential convergence and tractability will be an active research fieldin the future. References [1] N. Aronszajn, Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404,1950[2] J. Dick, P. Kritzer, F. Pillichshammer, H. Wo´zniakowski. Approximation of analyticfunctions in Korobov spaces. To appear in J. Complexity, 2014.[3] J. Dick, F.Y. Kuo, I.H. Sloan. High dimensional integration—the quasi-Monte Carloway. Acta Numer. 22, 133–288, 2013.[4] J. Dick, G. Larcher, F. Pillichshammer, H. Wo´zniakowski. Exponential convergenceand tractability of multivariate integration for Korobov spaces. Math. Comp. 80,905–930, 2011.[5] J. Dick, F. Pillichshammer.
Digital Nets and Sequences. Discrepancy Theory andQuasi-Monte Carlo Integration . Cambridge University Press, Cambridge, 2010.[6] M. Gnewuch, H. Wo´zniakowski, Generalized tractability for multivariate problems,Part II: Linear tensor product problems, linear information, unrestricted tractability.Found. Comput. Math. 9, 431–460, 2009.[7] M. Gnewuch, H. Wo´zniakowski. Quasi-polynomial tractability. J. Complexity 27,312–330, 2011.[8] P. Kritzer, F. Pillichshammer, H. Wo´zniakowski. Multivariate integration of infinitelymany times differentiable functions in weighted Korobov spaces. To appear in Math.Comp., 2014.[9] F.Y. Kuo, Ch. Schwab, I.H. Sloan. Quasi-Monte Carlo methods for high dimensionalintegration: the standard (weighted Hilbert space) setting and beyond. ANZIAM J.53, 1–37, 2011.[10] H. Niederreiter.
Random Number Generation and Quasi-Monte Carlo Methods .SIAM, Philadelphia, 1992.[11] E. Novak and H. Wo´zniakowski.
Tractability of Multivariate Problems, Volume I:Linear Information . EMS, Z¨urich, 2008.2212] E. Novak and H. Wo´zniakowski.
Tractability of Multivariate Problems, Volume II:Standard Information for Functionals . EMS, Z¨urich, 2010.[13] E. Novak and H. Wo´zniakowski.
Tractability of Multivariate Problems, Volume III:Standard Information for Operators . EMS, Z¨urich, 2012.[14] A. Papageorgiou and I. Petras, A new criterion for tractability of multivariate prob-lems. Submitted, 2013.[15] P. Siedlecki. Uniform weak tractability. J. Complexity 29, 438–453, 2013.[16] I.H. Sloan, H. Wo´zniakowski. When are quasi-Monte Carlo algorithms efficient forhigh dimensional integrals? J. Complexity 14, 1–33, 1998.[17] J.F. Traub, G.W. Wasilkowski, and H. Wo´zniakowski.